arXiv subjects

Vladimir Braverman

Publications and source records attributed to Vladimir Braverman.

At least 73 records · Page 4Linked to original sources

arXiv2022

Pretrained Models for Multilingual Federated Learning

Since the advent of Federated Learning (FL), research has applied these methods to natural language processing (NLP) tasks. Despite a plethora of papers in FL for NLP, no previous works have studied how multilingual text impacts FL algorithms. Furthermore, multilingual text provides an interesting avenue to examine the impact of non-IID text (e.g. different languages) on FL in naturally occurring data. We explore three multilingual language tasks, language modeling, machine translation, and text classification using differing federated and non-federated learning algorithms. Our results show that using pretrained models reduces the negative effects of FL, helping them to perform near or better than centralized (no privacy) learning, even when using non-IID partitioning.

cs.CL↗

arXiv2022

Sublinear Time Spectral Density Estimation

We present a new sublinear time algorithm for approximating the spectral density (eigenvalue distribution) of an $n\times n$ normalized graph adjacency or Laplacian matrix. The algorithm recovers the spectrum up to $ε$ accuracy in the Wasserstein-1 distance in $O(n\cdot \text{poly}(1/ε))$ time given sample access to the graph. This result compliments recent work by David Cohen-Steiner, Weihao Kong, Christian Sohler, and Gregory Valiant (2018), which obtains a solution with runtime independent of $n$, but exponential in $1/ε$. We conjecture that the trade-off between dimension dependence and accuracy is inherent. Our method is simple and works well experimentally. It is based on a Chebyshev polynomial moment matching method that employees randomized estimators for the matrix trace. We prove that, for any Hermitian $A$, this moment matching method returns an $ε$ approximation to the spectral density using just $O({1}/ε)$ matrix-vector products with $A$. By leveraging stability properties of the Chebyshev polynomial three-term recurrence, we then prove that the method is amenable to the use of coarse approximate matrix-vector products. Our sublinear time algorithm follows from combining this result with a novel sampling algorithm for approximating matrix-vector products with a normalized graph adjacency matrix. Of independent interest, we show a similar result for the widely used \emph{kernel polynomial method} (KPM), proving that this practical algorithm nearly matches the theoretical guarantees of our moment matching method. Our analysis uses tools from Jackson's seminal work on approximation with positive polynomial kernels.

cs.DS↗

arXiv2022

Gap-Dependent Unsupervised Exploration for Reinforcement Learning

For the problem of task-agnostic reinforcement learning (RL), an agent first collects samples from an unknown environment without the supervision of reward signals, then is revealed with a reward and is asked to compute a corresponding near-optimal policy. Existing approaches mainly concern the worst-case scenarios, in which no structural information of the reward/transition-dynamics is utilized. Therefore the best sample upper bound is $\propto\widetilde{\mathcal{O}}(1/ε^2)$, where $ε>0$ is the target accuracy of the obtained policy, and can be overly pessimistic. To tackle this issue, we provide an efficient algorithm that utilizes a gap parameter, $ρ>0$, to reduce the amount of exploration. In particular, for an unknown finite-horizon Markov decision process, the algorithm takes only $\widetilde{\mathcal{O}} (1/ε\cdot (H^3SA / ρ+ H^4 S^2 A) )$ episodes of exploration, and is able to obtain an $ε$-optimal policy for a post-revealed reward with sub-optimality gap at least $ρ$, where $S$ is the number of states, $A$ is the number of actions, and $H$ is the length of the horizon, obtaining a nearly \emph{quadratic saving} in terms of $ε$. We show that, information-theoretically, this bound is nearly tight for $ρ< Θ(1/(HS))$ and $H>1$. We further show that $\propto\widetilde{\mathcal{O}}(1)$ sample bound is possible for $H=1$ (i.e., multi-armed bandit) or with a sampling simulator, establishing a stark separation between those settings and the RL setting.

cs.LG↗

arXiv2022

New Coresets for Projective Clustering and Applications

$(j,k)$-projective clustering is the natural generalization of the family of $k$-clustering and $j$-subspace clustering problems. Given a set of points $P$ in $\mathbb{R}^d$, the goal is to find $k$ flats of dimension $j$, i.e., affine subspaces, that best fit $P$ under a given distance measure. In this paper, we propose the first algorithm that returns an $L_\infty$ coreset of size polynomial in $d$. Moreover, we give the first strong coreset construction for general $M$-estimator regression. Specifically, we show that our construction provides efficient coreset constructions for Cauchy, Welsch, Huber, Geman-McClure, Tukey, $L_1-L_2$, and Fair regression, as well as general concave and power-bounded loss functions. Finally, we provide experimental results based on real-world datasets, showing the efficacy of our approach.

cs.LG↗

arXiv2022

Risk Bounds of Multi-Pass SGD for Least Squares in the Interpolation Regime

Stochastic gradient descent (SGD) has achieved great success due to its superior performance in both optimization and generalization. Most of existing generalization analyses are made for single-pass SGD, which is a less practical variant compared to the commonly-used multi-pass SGD. Besides, theoretical analyses for multi-pass SGD often concern a worst-case instance in a class of problems, which may be pessimistic to explain the superior generalization ability for some particular problem instance. The goal of this paper is to sharply characterize the generalization of multi-pass SGD, by developing an instance-dependent excess risk bound for least squares in the interpolation regime, which is expressed as a function of the iteration number, stepsize, and data covariance. We show that the excess risk of SGD can be exactly decomposed into the excess risk of GD and a positive fluctuation error, suggesting that SGD always performs worse, instance-wisely, than GD, in generalization. On the other hand, we show that although SGD needs more iterations than GD to achieve the same level of excess risk, it saves the number of stochastic gradient evaluations, and therefore is preferable in terms of computational time.

cs.LG↗

arXiv2021

Cross-Domain Federated Learning in Medical Imaging

Federated learning is increasingly being explored in the field of medical imaging to train deep learning models on large scale datasets distributed across different data centers while preserving privacy by avoiding the need to transfer sensitive patient information. In this manuscript, we explore federated learning in a multi-domain, multi-task setting wherein different participating nodes may contain datasets sourced from different domains and are trained to solve different tasks. We evaluated cross-domain federated learning for the tasks of object detection and segmentation across two different experimental settings: multi-modal and multi-organ. The result from our experiments on cross-domain federated learning framework were very encouraging with an overlap similarity of 0.79 for organ localization and 0.65 for lesion segmentation. Our results demonstrate the potential of federated learning in developing multi-domain, multi-task deep learning models without sharing data from different domains.

eess.IV↗

arXiv2021

Coresets for Clustering with Missing Values

We provide the first coreset for clustering points in $\mathbb{R}^d$ that have multiple missing values (coordinates). Previous coreset constructions only allow one missing coordinate. The challenge in this setting is that objective functions, like $k$-Means, are evaluated only on the set of available (non-missing) coordinates, which varies across points. Recall that an $ε$-coreset of a large dataset is a small proxy, usually a reweighted subset of points, that $(1+ε)$-approximates the clustering objective for every possible center set. Our coresets for $k$-Means and $k$-Median clustering have size $(jk)^{O(\min(j,k))} (ε^{-1} d \log n)^2$, where $n$ is the number of data points, $d$ is the dimension and $j$ is the maximum number of missing coordinates for each data point. We further design an algorithm to construct these coresets in near-linear time, and consequently improve a recent quadratic-time PTAS for $k$-Means with missing values [Eiben et al., SODA 2021] to near-linear time. We validate our coreset construction, which is based on importance sampling and is easy to implement, on various real data sets. Our coreset exhibits a flexible tradeoff between coreset size and accuracy, and generally outperforms the uniform-sampling baseline. Furthermore, it significantly speeds up a Lloyd's-style heuristic for $k$-Means with missing values.

cs.DS↗

arXiv2021

Accommodating Picky Customers: Regret Bound and Exploration Complexity for Multi-Objective Reinforcement Learning

In this paper we consider multi-objective reinforcement learning where the objectives are balanced using preferences. In practice, the preferences are often given in an adversarial manner, e.g., customers can be picky in many applications. We formalize this problem as an episodic learning problem on a Markov decision process, where transitions are unknown and a reward function is the inner product of a preference vector with pre-specified multi-objective reward functions. We consider two settings. In the online setting, the agent receives a (adversarial) preference every episode and proposes policies to interact with the environment. We provide a model-based algorithm that achieves a nearly minimax optimal regret bound $\widetilde{\mathcal{O}}\bigl(\sqrt{\min\{d,S\}\cdot H^2 SAK}\bigr)$, where $d$ is the number of objectives, $S$ is the number of states, $A$ is the number of actions, $H$ is the length of the horizon, and $K$ is the number of episodes. Furthermore, we consider preference-free exploration, i.e., the agent first interacts with the environment without specifying any preference and then is able to accommodate arbitrary preference vector up to $ε$ error. Our proposed algorithm is provably efficient with a nearly optimal trajectory complexity $\widetilde{\mathcal{O}}\bigl({\min\{d,S\}\cdot H^3 SA}/{ε^2}\bigr)$. This result partly resolves an open problem raised by \citet{jin2020reward}.

cs.LG↗

arXiv2021

Adversarial Robustness of Streaming Algorithms through Importance Sampling

In this paper, we introduce adversarially robust streaming algorithms for central machine learning and algorithmic tasks, such as regression and clustering, as well as their more general counterparts, subspace embedding, low-rank approximation, and coreset construction. For regression and other numerical linear algebra related tasks, we consider the row arrival streaming model. Our results are based on a simple, but powerful, observation that many importance sampling-based algorithms give rise to adversarial robustness which is in contrast to sketching based algorithms, which are very prevalent in the streaming literature but suffer from adversarial attacks. In addition, we show that the well-known merge and reduce paradigm in streaming is adversarially robust. Since the merge and reduce paradigm allows coreset constructions in the streaming setting, we thus obtain robust algorithms for $k$-means, $k$-median, $k$-center, Bregman clustering, projective clustering, principal component analysis (PCA) and non-negative matrix factorization. To the best of our knowledge, these are the first adversarially robust results for these problems yet require no new algorithmic implementations. Finally, we empirically confirm the robustness of our algorithms on various adversarial attacks and demonstrate that by contrast, some common existing algorithms are not robust. (Abstract shortened to meet arXiv limits)

cs.LG↗

arXiv2021

Benign Overfitting of Constant-Stepsize SGD for Linear Regression

There is an increasing realization that algorithmic inductive biases are central in preventing overfitting; empirically, we often see a benign overfitting phenomenon in overparameterized settings for natural learning algorithms, such as stochastic gradient descent (SGD), where little to no explicit regularization has been employed. This work considers this issue in arguably the most basic setting: constant-stepsize SGD (with iterate averaging or tail averaging) for linear regression in the overparameterized regime. Our main result provides a sharp excess risk bound, stated in terms of the full eigenspectrum of the data covariance matrix, that reveals a bias-variance decomposition characterizing when generalization is possible: (i) the variance bound is characterized in terms of an effective dimension (specific for SGD) and (ii) the bias bound provides a sharp geometric characterization in terms of the location of the initial iterate (and how it aligns with the data covariance matrix). More specifically, for SGD with iterate averaging, we demonstrate the sharpness of the established excess risk bound by proving a matching lower bound (up to constant factors). For SGD with tail averaging, we show its advantage over SGD with iterate averaging by proving a better excess risk bound together with a nearly matching lower bound. Moreover, we reflect on a number of notable differences between the algorithmic regularization afforded by (unregularized) SGD in comparison to ordinary least squares (minimum-norm interpolation) and ridge regression. Experimental results on synthetic data corroborate our theoretical findings.

cs.LG↗

arXiv2021

Symmetric Norm Estimation and Regression on Sliding Windows

The sliding window model generalizes the standard streaming model and often performs better in applications where recent data is more important or more accurate than data that arrived prior to a certain time. We study the problem of approximating symmetric norms (a norm on $\mathbb{R}^n$ that is invariant under sign-flips and coordinate-wise permutations) in the sliding window model, where only the $W$ most recent updates define the underlying frequency vector. Whereas standard norm estimation algorithms for sliding windows rely on the smooth histogram framework of Braverman and Ostrovsky (FOCS 2007), analyzing the smoothness of general symmetric norms seems to be a challenging obstacle. Instead, we observe that the symmetric norm streaming algorithm of Braverman et. al. (STOC 2017) can be reduced to identifying and approximating the frequency of heavy-hitters in a number of substreams. We introduce a heavy-hitter algorithm that gives a $(1+ε)$-approximation to each of the reported frequencies in the sliding window model, thus obtaining the first algorithm for general symmetric norm estimation in the sliding window model. Our algorithm is a universal sketch that simultaneously approximates all symmetric norms in a parametrizable class and also improves upon the smooth histogram framework for estimating $L_p$ norms, for a range of large $p$. Finally, we consider the problem of overconstrained linear regression problem in the case that loss function that is an Orlicz norm, a symmetric norm that can be interpreted as a scale-invariant version of $M$-estimators. We give the first sublinear space algorithms that produce $(1+ε)$-approximate solutions to the linear regression problem for loss functions that are Orlicz norms in both the streaming and sliding window models.

cs.DS↗

arXiv2021

Near-Optimal Entrywise Sampling of Numerically Sparse Matrices

Many real-world data sets are sparse or almost sparse. One method to measure this for a matrix $A\in \mathbb{R}^{n\times n}$ is the \emph{numerical sparsity}, denoted $\mathsf{ns}(A)$, defined as the minimum $k\geq 1$ such that $\|a\|_1/\|a\|_2 \leq \sqrt{k}$ for every row and every column $a$ of $A$. This measure of $a$ is smooth and is clearly only smaller than the number of non-zeros in the row/column $a$. The seminal work of Achlioptas and McSherry [2007] has put forward the question of approximating an input matrix $A$ by entrywise sampling. More precisely, the goal is to quickly compute a sparse matrix $\tilde{A}$ satisfying $\|A - \tilde{A}\|_2 \leq ε\|A\|_2$ (i.e., additive spectral approximation) given an error parameter $ε>0$. The known schemes sample and rescale a small fraction of entries from $A$. We propose a scheme that sparsifies an almost-sparse matrix $A$ -- it produces a matrix $\tilde{A}$ with $O(ε^{-2}\mathsf{ns}(A) \cdot n\ln n)$ non-zero entries with high probability. We also prove that this upper bound on $\mathsf{nnz}(\tilde{A})$ is \emph{tight} up to logarithmic factors. Moreover, our upper bound improves when the spectrum of $A$ decays quickly (roughly replacing $n$ with the stable rank of $A$). Our scheme can be implemented in time $O(\mathsf{nnz}(A))$ when $\|A\|_2$ is given. Previously, a similar upper bound was obtained by Achlioptas et. al [2013] but only for a restricted class of inputs that does not even include symmetric or covariance matrices. Finally, we demonstrate two applications of these sampling techniques, to faster approximate matrix multiplication, and to ridge regression by using sparse preconditioners.

cs.DS↗

arXiv2021

Lifelong Learning with Sketched Structural Regularization

Preventing catastrophic forgetting while continually learning new tasks is an essential problem in lifelong learning. Structural regularization (SR) refers to a family of algorithms that mitigate catastrophic forgetting by penalizing the network for changing its "critical parameters" from previous tasks while learning a new one. The penalty is often induced via a quadratic regularizer defined by an \emph{importance matrix}, e.g., the (empirical) Fisher information matrix in the Elastic Weight Consolidation framework. In practice and due to computational constraints, most SR methods crudely approximate the importance matrix by its diagonal. In this paper, we propose \emph{Sketched Structural Regularization} (Sketched SR) as an alternative approach to compress the importance matrices used for regularizing in SR methods. Specifically, we apply \emph{linear sketching methods} to better approximate the importance matrices in SR algorithms. We show that sketched SR: (i) is computationally efficient and straightforward to implement, (ii) provides an approximation error that is justified in theory, and (iii) is method oblivious by construction and can be adapted to any method that belongs to the structural regularization class. We show that our proposed approach consistently improves various SR algorithms' performance on both synthetic experiments and benchmark continual learning tasks, including permuted-MNIST and CIFAR-100.

cs.LG↗

arXiv2021

Direction Matters: On the Implicit Bias of Stochastic Gradient Descent with Moderate Learning Rate

Understanding the algorithmic bias of \emph{stochastic gradient descent} (SGD) is one of the key challenges in modern machine learning and deep learning theory. Most of the existing works, however, focus on \emph{very small or even infinitesimal} learning rate regime, and fail to cover practical scenarios where the learning rate is \emph{moderate and annealing}. In this paper, we make an initial attempt to characterize the particular regularization effect of SGD in the moderate learning rate regime by studying its behavior for optimizing an overparameterized linear regression problem. In this case, SGD and GD are known to converge to the unique minimum-norm solution; however, with the moderate and annealing learning rate, we show that they exhibit different \emph{directional bias}: SGD converges along the large eigenvalue directions of the data matrix, while GD goes after the small eigenvalue directions. Furthermore, we show that such directional bias does matter when early stopping is adopted, where the SGD output is nearly optimal but the GD output is suboptimal. Finally, our theory explains several folk arts in practice used for SGD hyperparameter tuning, such as (1) linearly scaling the initial learning rate with batch size; and (2) overrunning SGD with high learning rate even when the loss stops decreasing.

cs.LG↗

arXiv2020

Sparse Coresets for SVD on Infinite Streams

In streaming Singular Value Decomposition (SVD), $d$-dimensional rows of a possibly infinite matrix arrive sequentially as points in $\mathbb{R}^d$. An $ε$-coreset is a (much smaller) matrix whose sum of square distances of the rows to any hyperplane approximates that of the original matrix to a $1 \pm ε$ factor. Our main result is that we can maintain a $ε$-coreset while storing only $O(d \log^2 d / ε^2)$ rows. Known lower bounds of $Ω(d / ε^2)$ rows show that this is nearly optimal. Moreover, each row of our coreset is a weighted subset of the input rows. This is highly desirable since it: (1) preserves sparsity; (2) is easily interpretable; (3) avoids precision errors; (4) applies to problems with constraints on the input. Previous streaming results for SVD that return a subset of the input required storing $Ω(d \log^3 n / ε^2)$ rows where $n$ is the number of rows seen so far. Our algorithm, with storage independent of $n$, is the first result that uses finite memory on infinite streams. We support our findings with experiments on the Wikipedia dataset benchmarked against state-of-the-art algorithms.

cs.DS↗

arXiv2020

Sketch and Scale: Geo-distributed tSNE and UMAP

Running machine learning analytics over geographically distributed datasets is a rapidly arising problem in the world of data management policies ensuring privacy and data security. Visualizing high dimensional data using tools such as t-distributed Stochastic Neighbor Embedding (tSNE) and Uniform Manifold Approximation and Projection (UMAP) became common practice for data scientists. Both tools scale poorly in time and memory. While recent optimizations showed successful handling of 10,000 data points, scaling beyond million points is still challenging. We introduce a novel framework: Sketch and Scale (SnS). It leverages a Count Sketch data structure to compress the data on the edge nodes, aggregates the reduced size sketches on the master node, and runs vanilla tSNE or UMAP on the summary, representing the densest areas, extracted from the aggregated sketch. We show this technique to be fully parallel, scale linearly in time, logarithmically in memory, and communication, making it possible to analyze datasets with many millions, potentially billions of data points, spread across several data centers around the globe. We demonstrate the power of our method on two mid-size datasets: cancer data with 52 million 35-band pixels from multiple images of tumor biopsies; and astrophysics data of 100 million stars with multi-color photometry from the Sloan Digital Sky Survey (SDSS).

cs.DC↗

arXiv2020

FetchSGD: Communication-Efficient Federated Learning with Sketching

Existing approaches to federated learning suffer from a communication bottleneck as well as convergence issues due to sparse client participation. In this paper we introduce a novel algorithm, called FetchSGD, to overcome these challenges. FetchSGD compresses model updates using a Count Sketch, and then takes advantage of the mergeability of sketches to combine model updates from many workers. A key insight in the design of FetchSGD is that, because the Count Sketch is linear, momentum and error accumulation can both be carried out within the sketch. This allows the algorithm to move momentum and error accumulation from clients to the central aggregator, overcoming the challenges of sparse client participation while still achieving high compression rates and good convergence. We prove that FetchSGD has favorable convergence guarantees, and we demonstrate its empirical effectiveness by training two residual networks and a transformer model.

cs.LG↗

arXiv2020

Data-Independent Structured Pruning of Neural Networks via Coresets

Model compression is crucial for deployment of neural networks on devices with limited computational and memory resources. Many different methods show comparable accuracy of the compressed model and similar compression rates. However, the majority of the compression methods are based on heuristics and offer no worst-case guarantees on the trade-off between the compression rate and the approximation error for an arbitrarily new sample. We propose the first efficient structured pruning algorithm with a provable trade-off between its compression rate and the approximation error for any future test sample. Our method is based on the coreset framework and it approximates the output of a layer of neurons/filters by a coreset of neurons/filters in the previous layer and discards the rest. We apply this framework in a layer-by-layer fashion from the bottom to the top. Unlike previous works, our coreset is data independent, meaning that it provably guarantees the accuracy of the function for any input $x\in \mathbb{R}^d$, including an adversarial one.

cs.LG↗